Denote by \(D\)  the (dis-)similarity between the morphed moving image \(I\circ \Phi \)  and the fixed image \(J\) . Pixel-based metrics are well explored within algorithmic image registration, a comparative evaluation is given by [1]}. We briefly recall the two most popular choices, mean squared error (\(\operatorname{MSE}\) ) and normalized cross correlation (\(\operatorname{NCC}\) ).
The pixel-wise MSE is intuitive and computationally efficient. It is derived from maximizing the negative log-likelihood of a Gaussian normal distribution, making it an appropriate choice under the assumption of Gaussian noise. On a grid of discrete points \(\mathbf {p}\)  from domain \(\Omega \) , the \(\operatorname{MSE}\)  is defined as \(\operatorname{MSE}(I\circ \Phi , J) = \frac{1}{|\Omega |} \sum _{\mathbf {p}\in \Omega } \Vert I\circ \Phi (\mathbf {p}) - J(\mathbf {p}) \Vert ^2\) .
